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Further Maths

  • A-Level Physics

Capacitors

Updated: Aug 22

Capacitors store electrical charge. They are made up of two conducting plates separated by a dielectric (insulator) or gap. A positive and a negative charge builds up on the opposite plates, but the insulator stops the charge from moving – this creates a potential difference via a uniform electric field. The charges on each plate are equal and opposite.

Capacitance is the of a capacitor is defined as the charge stored per unit p.d. across the capacitor.

C =Q/V capacitance = charge / potential difference

The units of capacitance are Farads, F, or Coulomb per Volt, C/V.



Capacitor Networks

As a general rule of thumb, everything about capacitors is the opposite to how it would be with resistors:

1/C = 1/C(1) + 1/C(2) … Combined Capacitance in series
C = C(1) + C(2) … Combined Capacitance in parallel


Energy Stored

Work is done removing negative charge from one plate and depositing it on the other plate – this is the same as the energy stored by the capacitor, and can be found as the area under a p.d. – charge graph:

W = 1/2 QV Energy stored = 1/2 x charge x p.d.

This equation can be rewritten using the capacitance equation, C = Q/V, rearranged as Q = CV

W = 1/2 V²C Energy Stored = 1/2 voltage² x capacitance

Or, using the capacitance equation written in the form V = Q/C:

W = 1/2 Q²/C Energy stored = 1/2 charge² / capacitance

Uses for Capacitors as Energy Stores

  • Flash Photography Before LEDs, you would need a short burst of high current to give a bright flash. A capacitor discharging quickly achieved this.

  • Back-up Power Supplies If there is a power outage, many large capacitors can be used to keep key systems running for a short period of time.

  • Smoothing out p.d. When AC is converted to DC, capacitors charge up at the peaks and discharge at the troughs to supply a constant, smooth power output.



Charging & Discharging

The p.d. and charge across a capacitor increase over time as the capacitor is charged, but the current reduces – when this reaches 0 A, the capacitor is fully charged and the p.d. across it is equal to that of the power supply, V(0).

To measure capacitance, p.d. and current across a charging capacitor, you need to set up a circuit where the capacitor is in series to a resistor. To measure discharge, however, the resistor and capacitor must be in parallel.

As soon as the capacitor is connected to the power supply, electrons flow onto the plate connected to the negative terminal. This creates a negative charge on one of the plates, so the electrons on the other plate are repelled towards the positive terminal, leaving that plate positive. An equal but opposite charge builds on each plate, creating a p.d. between them. To begin with, the current is high, but it drops as electrostatic repulsion makes it harder for electrons to be deposited.

For discharging, the Charge and p.d. graphs are the opposite (or negative), but the current graph is the same. The capacitor is fully discharged when both the current and p.d. are 0. 



Time Constant & Exponential Decay

The time taken for a capacitor to charge or discharge depends on two things:

  • the capacitance, as this affects the amount of charge that can be transferred at a given voltage

  • the resistance, as this affects the current in the circuit

However, the pattern of charging/discharging is always the same: exponential decay. This means that for a set period of time, the charge, p.d. or current decreases by the same ratio. This is modelled with the number e:

C is the capacitance of the capacitor, and R the resistance in the circuit. Because these are constant, they are known as the time constant, τ:

τ = CR time constant = capacitance x resistance

The time constant is the time taken for the charge, p.d. or current to fall to 37% (1/e) of its initial value, or for the p.d. or charge to rise to 63% of its maximum value. The larger the resistor in series with the capacitor, the longer it takes to charge/discharge.

The time for the capacitor to charge/discharge fully is around 5τ.

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